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Properties

Label 15.5.33005951466...0999.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,11^{12}\cdot 19^{5}\cdot 461^{13}$
Root discriminant $3697.48$
Ramified primes $11, 19, 461$
Class number $325024400$ (GRH)
Class group $[2, 2, 10, 8125610]$ (GRH)
Galois group $S_3 \times C_5$ (as 15T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![543122152076214272, 322453908130652160, 74237750960369664, 11110764946271360, 7140439214752, -215344958988368, -14585961566768, 1679913052736, 127789199194, -9981552889, -125770086, 12591603, 37278, -5995, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 5995*x^13 + 37278*x^12 + 12591603*x^11 - 125770086*x^10 - 9981552889*x^9 + 127789199194*x^8 + 1679913052736*x^7 - 14585961566768*x^6 - 215344958988368*x^5 + 7140439214752*x^4 + 11110764946271360*x^3 + 74237750960369664*x^2 + 322453908130652160*x + 543122152076214272)
 
gp: K = bnfinit(x^15 - 2*x^14 - 5995*x^13 + 37278*x^12 + 12591603*x^11 - 125770086*x^10 - 9981552889*x^9 + 127789199194*x^8 + 1679913052736*x^7 - 14585961566768*x^6 - 215344958988368*x^5 + 7140439214752*x^4 + 11110764946271360*x^3 + 74237750960369664*x^2 + 322453908130652160*x + 543122152076214272, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 5995 x^{13} + 37278 x^{12} + 12591603 x^{11} - 125770086 x^{10} - 9981552889 x^{9} + 127789199194 x^{8} + 1679913052736 x^{7} - 14585961566768 x^{6} - 215344958988368 x^{5} + 7140439214752 x^{4} + 11110764946271360 x^{3} + 74237750960369664 x^{2} + 322453908130652160 x + 543122152076214272 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-330059514666048497176686804227168786358108620765840999=-\,11^{12}\cdot 19^{5}\cdot 461^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3697.48$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 19, 461$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{118016} a^{10} + \frac{259}{118016} a^{9} + \frac{213}{59008} a^{8} + \frac{1523}{59008} a^{7} - \frac{3199}{118016} a^{6} + \frac{5395}{118016} a^{5} + \frac{77}{29504} a^{4} - \frac{3055}{29504} a^{3} + \frac{345}{3688} a^{2} - \frac{903}{1844} a - \frac{75}{461}$, $\frac{1}{472064} a^{11} - \frac{1}{472064} a^{10} - \frac{363}{118016} a^{9} - \frac{381}{236032} a^{8} + \frac{6981}{472064} a^{7} - \frac{3729}{472064} a^{6} + \frac{13815}{236032} a^{5} - \frac{5557}{118016} a^{4} - \frac{14065}{59008} a^{3} - \frac{2533}{14752} a^{2} + \frac{73}{1844} a - \frac{196}{461}$, $\frac{1}{61084137472} a^{12} + \frac{12485}{15271034368} a^{11} - \frac{132653}{61084137472} a^{10} + \frac{67940055}{30542068736} a^{9} - \frac{18340581}{61084137472} a^{8} - \frac{46106951}{1908879296} a^{7} + \frac{826439677}{61084137472} a^{6} - \frac{1475157893}{30542068736} a^{5} + \frac{329597469}{15271034368} a^{4} - \frac{1836929691}{7635517184} a^{3} - \frac{232110827}{1908879296} a^{2} + \frac{33097783}{119304956} a - \frac{5451096}{29826239}$, $\frac{1}{244336549888} a^{13} + \frac{10197}{244336549888} a^{11} - \frac{10797}{30542068736} a^{10} - \frac{901396557}{244336549888} a^{9} + \frac{16364343}{7635517184} a^{8} + \frac{5106115175}{244336549888} a^{7} - \frac{738540705}{30542068736} a^{6} - \frac{228728761}{3817758592} a^{5} + \frac{726303621}{15271034368} a^{4} - \frac{331401563}{15271034368} a^{3} + \frac{734581789}{3817758592} a^{2} + \frac{3892443}{20748688} a + \frac{1220042}{29826239}$, $\frac{1}{3267992289395867071175296290395688739802231342162918530918966088538569854910464} a^{14} + \frac{683099048027881183857689741703542273893241898016127859338398552301}{816998072348966767793824072598922184950557835540729632729741522134642463727616} a^{13} + \frac{179739989834737730176187171794481230453187053209045349280365561611}{142086621278081177007621577843290814774010058354909501344302873414720428474368} a^{12} - \frac{765382156267385384092642165588686645883157349147516014485707087953427833}{816998072348966767793824072598922184950557835540729632729741522134642463727616} a^{11} - \frac{8563213276941862631120604398480647288083232183955010133383679846071145}{7088920367453073907104764187409303123215252369116959936917496938261539815424} a^{10} + \frac{2013417192193680298884939354284594502185964154841566887549270548718227594699}{816998072348966767793824072598922184950557835540729632729741522134642463727616} a^{9} - \frac{7404983653578909097605914368061995255174108961077956077178469759021799828641}{3267992289395867071175296290395688739802231342162918530918966088538569854910464} a^{8} - \frac{15717267830460409201575102005821390441668144675131168763052896339808878343231}{816998072348966767793824072598922184950557835540729632729741522134642463727616} a^{7} + \frac{706913186176809277762669258040761939423072744210815872753580981448312482401}{408499036174483383896912036299461092475278917770364816364870761067321231863808} a^{6} + \frac{1014417113848523987357098568833381391183617641157187843696182355676216643585}{51062379521810422987114004537432636559409864721295602045608845133415153982976} a^{5} + \frac{11701641363750530381455769661594550950182352212577497754399331716279271158683}{204249518087241691948456018149730546237639458885182408182435380533660615931904} a^{4} - \frac{494283776260908406893149970631505911545253685971626706332215070981446977041}{51062379521810422987114004537432636559409864721295602045608845133415153982976} a^{3} + \frac{173865714174288664708002434965710706821328863638297558962274865983322448489}{6382797440226302873389250567179079569926233090161950255701105641676894247872} a^{2} + \frac{87019729496291960336572399934424848654554520149266405644267594670246824421}{797849680028287859173656320897384946240779136270243781962638205209611780984} a + \frac{4029893818031590104572999718575623074163641989956007874751047289811940669}{99731210003535982396707040112173118280097392033780472745329775651201472623}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{8125610}$, which has order $325024400$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 62905376746024980 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_3$ (as 15T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.1.8759.1, 5.5.661263333631681.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{15}$ $15$ $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$19$19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
461Data not computed